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Mirrors > Home > MPE Home > Th. List > oppgsubg | Structured version Visualization version GIF version |
Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
Ref | Expression |
---|---|
oppggic.o | ⊢ 𝑂 = (oppg‘𝐺) |
Ref | Expression |
---|---|
oppgsubg | ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 19070 | . . 3 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
2 | subgrcl 19070 | . . . 4 ⊢ (𝑥 ∈ (SubGrp‘𝑂) → 𝑂 ∈ Grp) | |
3 | oppggic.o | . . . . 5 ⊢ 𝑂 = (oppg‘𝐺) | |
4 | 3 | oppggrpb 19296 | . . . 4 ⊢ (𝐺 ∈ Grp ↔ 𝑂 ∈ Grp) |
5 | 2, 4 | sylibr 233 | . . 3 ⊢ (𝑥 ∈ (SubGrp‘𝑂) → 𝐺 ∈ Grp) |
6 | 3 | oppgsubm 19300 | . . . . . . 7 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂) |
7 | 6 | eleq2i 2820 | . . . . . 6 ⊢ (𝑥 ∈ (SubMnd‘𝐺) ↔ 𝑥 ∈ (SubMnd‘𝑂)) |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubMnd‘𝐺) ↔ 𝑥 ∈ (SubMnd‘𝑂))) |
9 | eqid 2727 | . . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | 3, 9 | oppginv 19297 | . . . . . . . 8 ⊢ (𝐺 ∈ Grp → (invg‘𝐺) = (invg‘𝑂)) |
11 | 10 | fveq1d 6893 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘𝑦) = ((invg‘𝑂)‘𝑦)) |
12 | 11 | eleq1d 2813 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (((invg‘𝐺)‘𝑦) ∈ 𝑥 ↔ ((invg‘𝑂)‘𝑦) ∈ 𝑥)) |
13 | 12 | ralbidv 3172 | . . . . 5 ⊢ (𝐺 ∈ Grp → (∀𝑦 ∈ 𝑥 ((invg‘𝐺)‘𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥)) |
14 | 8, 13 | anbi12d 630 | . . . 4 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ (SubMnd‘𝐺) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝐺)‘𝑦) ∈ 𝑥) ↔ (𝑥 ∈ (SubMnd‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥))) |
15 | 9 | issubg3 19083 | . . . 4 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubGrp‘𝐺) ↔ (𝑥 ∈ (SubMnd‘𝐺) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝐺)‘𝑦) ∈ 𝑥))) |
16 | eqid 2727 | . . . . . 6 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
17 | 16 | issubg3 19083 | . . . . 5 ⊢ (𝑂 ∈ Grp → (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑥 ∈ (SubMnd‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥))) |
18 | 4, 17 | sylbi 216 | . . . 4 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑥 ∈ (SubMnd‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥))) |
19 | 14, 15, 18 | 3bitr4d 311 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝑂))) |
20 | 1, 5, 19 | pm5.21nii 378 | . 2 ⊢ (𝑥 ∈ (SubGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝑂)) |
21 | 20 | eqriv 2724 | 1 ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ‘cfv 6542 SubMndcsubmnd 18724 Grpcgrp 18875 invgcminusg 18876 SubGrpcsubg 19059 oppgcoppg 19280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-grp 18878 df-minusg 18879 df-subg 19062 df-oppg 19281 |
This theorem is referenced by: lsmmod2 19615 lsmdisj2r 19624 |
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